This study investigated the unitary equivalent classes of one-dimensional quantum walks. We determined the unitary equivalent classes of one-dimensional quantum walks, two-phase quantum walks with one defect, complete two-phase quantum walks, one-dimensional quantum walks with one defect and translation-invariant one-dimensional quantum walks. The unitary equivalent classes of one-dimensional quantum walks with initial states were also considered.

arXiv:1704.07976Apr 26, 20170 citations

Physical Review A

One-dimensional quantum walk with unitary noise (8 pages)

Daniel Shapira

Published Jan 1, 2003• 0 citations

OpenAlex

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The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin $(n=0)$ at time $t=0,$ the position distribution ${P}_{t}(n)$ at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\ensuremath{\sigma}(t)$ scales as $\ensuremath{\sigma}(t)\ensuremath{\sim}t,$ unlike the classical random walk for which $\ensuremath{\sigma}(t)\ensuremath{\sim}\sqrt{t}.$ It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be $T\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}2},$ where $\ensuremath{\alpha}$ is the standard deviation of the noise.The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin $(n=0)$ at time $t=0,$ the position distribution ${P}_{t}(n)$ at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\ensuremath{\sigma}(t)$ scales as $\ensuremath{\sigma}(t)\ensuremath{\sim}t,$ unlike the classical random walk for which $\ensuremath{\sigma}(t)\ensuremath{\sim}\sqrt{t}.$ It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be $T\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}2},$ where $\ensuremath{\alpha}$ is the standard deviation of the noise.

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Unitary equivalent classes of one-dimensional quantum walks II

Hiromichi Ohno

arXiv:1704.07976Apr 26, 20170 citations

The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin

(n=0)

at time

t=0,

the position distribution

{P}_{t}(n)

at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\ensuremath{\sigma}(t)$ scales as $\ensuremath{\sigma}(t)\ensuremath{\sim}t,$ unlike the classical random walk for which $\ensuremath{\sigma}(t)\ensuremath{\sim}\sqrt{t}.$ It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be $T\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}2},$ where $\ensuremath{\alpha}$ is the standard deviation of the noise.